Definition:Series/Number Field

Definition

Let $S$ be one of the standard number fields $\R$, or $\C$.

Let $\sequence {a_n}$ be a sequence in $S$.

The series is what results when $\sequence {a_n}$ is summed to infinity:

$\ds \sum_{n \mathop = 1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$

A real series $S_n$ is the limit to infinity of the sequence of partial sums of a real sequence $\sequence {a_n}$:

$\ds S_n$

$=$

$\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty a_n$

$\ds $

$=$

$\ds a_1 + a_2 + a_3 + \cdots$

A complex series $S_n$ is the limit to infinity of the sequence of partial sums of a complex sequence $\sequence {a_n}$:

$\ds S_n$

$=$

$\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty a_n$

$\ds $

$=$

$\ds a_1 + a_2 + a_3 + \cdots$

Much of the original work on series of real and complex numbers was done by Leonhard Paul Euler.

The main bulk of the work to placed the concept on a rigorous footing was done by Carl Friedrich Gauss, Niels Henrik Abel‎ and Augustin Louis Cauchy.